We may proceed via arguments similar to those applied in the proof of corresponding results for ODEs [1]}. We assume the same local conditions as in the proof of Proposition REF , in particular that the \((n-k) \times (n-k)\)  matrix \(D_yf|_S\)  is regular so that \(S\)  is given locally as a graph \(y = \varphi _0(x)\) . Due to normal hyperbolicity, the nonlinear fast fibers \(\mathcal {F}\)  are given as constant level sets \(L(x,y) = c\) , for a \(C^r-\) smooth function \(L:\mathbb {R}^k \times \mathbb {R}^{n-k} \rightarrow \mathbb {R}^{n-k}\)  such that the \((n-k) \times k\)  matrix \(D_xL\)  is locally regular and the following invariance property is satisfied:
\(L(x + N^x(x,y) f(x,y), y + N^y(x,y) f(x,y) ) = L(x,y) .\) 
